1 Introduction

By \({\mathbb {C}},{\mathbb {R}},{\mathbb {Z}},{\mathbb {N}}_{0},{\mathbb {N}}_{1} ,{\mathbb {N}}_{2}\) let us denote the sets of complex numbers, real numbers, all integers, nonnegative integers, positive integers and the integers not smaller than 2,  respectively. We say that a domain \({\mathcal {G}}\subset {\mathbb {C}} ^{n},n\in {\mathbb {N}}_{1},\) is complete n-circular if \(z\lambda =(z_{1} \lambda _{1},...,z_{n}\lambda _{n})\in {\mathcal {G}}\) for each \(z=(z_{1} ,...,z_{n})\in {\mathcal {G}}\) and every \(\lambda =(\lambda _{1},...,\lambda _{n} )\in \overline{U^{n}}\), where U is the unit disc \(\{\zeta \in {\mathbb {C}} :|\zeta |<1\}\). From now on by \({\mathcal {G}}\) will be denoted a bounded complete n-circular domain in \({\mathbb {C}}^{n},n\in {\mathbb {N}}_{1}.\) Of course, only the open discs with the centre \(\zeta =0\) and the radius \(r>0,\) are the bounded complete 1-circular domains \({\mathcal {G}}\subset \) \({\mathbb {C}}.\)

In our considerations the Minkowski function \(\mu _{{\mathcal {G}}} :{\mathbb {C}}^{n}\rightarrow [0,\infty )\)

$$\begin{aligned} \mu _{{\mathcal {G}}}(z)=inf\left\{t>0:\frac{1}{t}z\in {\mathcal {G}}\right\},z\in {\mathbb {C}}^{n}, \end{aligned}$$

will be very useful. It is known (see e.g, [26]) that \(\mu _{{\mathcal {G}}}\) is a norm in \({\mathbb {C}}^{n}\) if \({\mathcal {G}}\) is a convex bounded complete n-circular domain. This function gives the possibility to redefine the domain \({\mathcal {G}}\) and its boundary \(\partial {\mathcal {G}}\) as follows:

$$\begin{aligned} {\mathcal {G}}=\{z\in {\mathbb {C}}^{n}:\mu _{{\mathcal {G}}}(z)<1\},\partial {\mathcal {G}}=\{z\in {\mathbb {C}}^{n}:\mu _{{\mathcal {G}}}(z)=1\}. \end{aligned}$$

Now, we recall some information about m-homogeneous polynomials. We say that a function \(Q_{m}:{\mathbb {C}}^{n}\longrightarrow {\mathbb {C}},m\in {\mathbb {N}}_{1},\) is an m-homogeneous polynomial if

$$\begin{aligned} Q_{m}(z)=L_{m}(z^{m})=L_{m}(z,...,z),z\in {\mathbb {C}}^{n}, \end{aligned}$$

where \(L_{m}:\left( {\mathbb {C}}^{n}\right) ^{m}\longrightarrow {\mathbb {C}}\) is a bounded m-linear function (by \(Q_{0}\) we note a complex constant). For this reason it is very natural to define (see [4]) the following generalization of the norm of m-homogeneous polynomials \(Q_{m} :{\mathbb {C}}^{n}\rightarrow {\mathbb {C}},\) i.e., the \(\mu _{{\mathcal {G}}}\)-balance of such m-homogeneous polynomials

$$\begin{aligned} \mu _{{\mathcal {G}}}(Q_{m})=\sup _{w\in {\mathbb {C}}^{n}\setminus \{0\}} \frac{\left| Q_{m}(w)\right| }{(\mu _{G}(w))^{m}}=\sup _{v\in \partial {\mathcal {G}}}\left| Q_{m}(v)\right| =\sup _{u\in {\mathcal {G}} }\left| Q_{m}(u)\right| . \end{aligned}$$

A simple kind of 1-homogeneous polynomials are the linear functionals \(J,I\in \left( {\mathbb {C}}^{n}\right) ^{*}\) of the form

$$\begin{aligned} J(z)=\sum \limits _{j=1}^{n}z_{j},z=(z_{1},...,z_{n})\in {\mathbb {C}} ^{n},I(z)=\left( \mu _{{\mathcal {G}}}(J)\right) ^{-1}J(z). \end{aligned}$$

Note that for \(m\in {\mathbb {N}}_{1},\) the mapping

$$\begin{aligned} I^{m}:{\mathbb {C}}^{n}\rightarrow {\mathbb {C}},I^{m}(z)=\left( I(z)\right) ^{m},z\in {\mathbb {C}}^{n}, \end{aligned}$$

is an m-homogeneous polynomial and \(\mu _{{\mathcal {G}}}(I^{m})=1.\)

By \({\mathcal {F}}({\mathcal {G}},{\mathbb {C}}^{m}),m\in {\mathbb {N}}_{1},\) let us denote the space of all functions \(f:{\mathcal {G}}\longrightarrow {\mathbb {C}}^{m},\) by \({\mathcal {H}}_{{\mathcal {G}}}\) the space of all holomorphic functions \(f\in {\mathcal {F}}({\mathcal {G}},{\mathbb {C}})\) and by \({\mathcal {H}}_{{\mathcal {G}} }(0),{\mathcal {H}}_{{\mathcal {G}}}(1)\) the collection of all \(f\in {\mathcal {H}}_{\mathcal{G}} \), normalized by \(f(0)=0,f(0)=1,\) respectively. Let us recall that every function \(f\in {\mathcal {H}}_{{\mathcal {G}}}\) has a unique power series expansion

$$\begin{aligned} f(z)=\sum _{m=0}^{\infty }Q_{f,m}(z),z\in {\mathcal {G}}, \end{aligned}$$
(1.1)

where \(Q_{f,m}:{\mathbb {C}}^{n}\rightarrow {\mathbb {C}},m\in {\mathbb {N}}_{0},\) are m-homogeneous polynomials determined uniquelly by the mth Frechèt differential \(D^{m}f(0)\) of f at zero via the formula

$$\begin{aligned} Q_{f,m}(z)=\frac{1}{m!}D^{m}f(0)(z^{m}). \end{aligned}$$

Many authors considered some problems connected with m-homogeneous polynomials in the power series expansion (1.1) of functions from different subfamilies of \({\mathcal {H}}_{{\mathcal {G}}}\) (see for instance [2, 5, 9, 13, 24]). In particular, in the case \({\mathcal{G}} \subset {\mathbb{C}}^{2}\) Bavrin [2] gives the following sharp estimates

$$\begin{aligned} \sup _{z\in {\mathcal {G}}}\left| Q_{f,m}(z)\right| \le 1-\left| Q_{f,0}\right| ^{2},m\in {\mathbb {N}}_{1}, \end{aligned}$$

for the homogeneous polynomials \(Q_{f,m}\) of functions belonging to the family

$$\begin{aligned} {\mathcal {B}}_{{\mathcal {G}}}=\left\{ f\in {\mathcal {H}}_{{\mathcal {G}}}:\left| f(z)\right| <1,z\in {\mathcal {G}}\right\} . \end{aligned}$$

Note, that the sharpness in the Bavrin’s result was understood as follows: There exists a bounded complete 2-circular domain \({\mathcal {G}}\subset \) \({\mathbb {C}}^{2}\) and a function \(f\in {\mathcal {B}}_{{\mathcal {G}}}\) which realizes the equality in the above inequality. Let us observe that in general case \({\mathcal{G}} {\subset }{\mathbb {C}}^{n},n\in {\mathbb {N}}_{1},\) the above inequality takes currently the following form

$$\begin{aligned} \mu _{{\mathcal {G}}}\left( Q_{f,m}\right) \le 1-\left| Q_{f,0}\right| ^{2},m\in {\mathbb {N}}_{1}, \end{aligned}$$

by the definition of the \(\mu _{{\mathcal {G}}}\)-balance \(\mu _{{\mathcal {G}}} (Q_{m}).\)

In the paper, we solve for \(\lambda \in {\mathbb {C}},k\in {\mathbb {N}}_{2},\) the problem of the sharp upper estimate

$$\begin{aligned} \mu _{{\mathcal {G}}}\left(Q_{f,2k}-\lambda \left( Q_{f,k}\right) ^{2}\right)\le M(\lambda ,k) \end{aligned}$$

for the pairs \(Q_{f,k},Q_{f,2k}\) homogeneous polynomials of functions belonging to some Bavrin’s subfamilies of the families \({\mathcal {H}} _{{\mathcal {G}}}(0),{\mathcal {H}}_{{\mathcal {G}}}(1)\) (the case \(k=1\) see the section Final Remarks). Moreover, here the sharpness is understand more generally. It means that for every bounded complete n-circular domain \({\mathcal {G}}\subset \) \({\mathbb {C}}^{n}\) there exists a function f,  which belongs to a mentioned Bavrin’s subfamily and realizes the equality in the above inequality.

Note that the afore-mentioned estimate is a generealization of the well known planar Fekete–Szegö [8] result onto the s.c.v. case.

In the sequel we use a special kind of functions symmetry. Let us observe that bounded complete n-circular domains \({\mathcal {G}}\subset {\mathbb {C}}^{n}\) are k-symmetric sets, \(k\in {\mathbb {N}}_{2},\) that is \(\varepsilon \mathcal {G}= \mathcal{G},\) where \(\varepsilon =\varepsilon _{k}=\exp \frac{2\pi i}{k}\) is a generator of the cyclic group of kth roots of unity. For \(k\in {\mathbb {N}}_{2},j\in {\mathbb {Z}}\) we define the collections \({\mathcal {F}}_{j,k}({\mathcal {G}},{\mathbb {C}}^{m})\) of functions \(f\in {\mathcal {F}}({\mathcal {G}},{\mathbb {C}}^{m}),\left( j,k\right) \)-symmetrical, i.e.,

$$\begin{aligned} f\left( \varepsilon z\right) =\varepsilon ^{j}f\left( z\right) ,\text { }z\in {\mathcal {G}}. \end{aligned}$$

Let us observe that \({\mathcal {F}}_{j,k}({\mathcal {G}},{\mathbb {C}}^{m} )\ne {\mathcal {F}}_{l,k}({\mathcal {G}},{\mathbb {C}}^{m})\) for different \(j,l\in \left\{ 0,1,...,k-1\right\} .\) Moreover, the intersections \({\mathcal {F}}_{j,k}({\mathcal {G}},{\mathbb {C}}^{m})\cap {\mathcal {F}}_{l,k} ({\mathcal {G}},{\mathbb {C}}^{m})\) are the singleton \(\left\{ 0\right\} \) for such j, l.

Now we present a functions decomposition theorem [19].

Theorem A

For every function \(f\in {\mathcal {F}}({\mathcal {G}},{\mathbb {C}}^{m})\) and every \(k\in {\mathbb {N}}_{2}\) there exists exactly one sequence of functions \(f_{j,k}\in {\mathcal {F}}_{j,k}({\mathcal {G}},{\mathbb {C}}^{m}),\) \(j=0,1,...,k-1,\) such that

$$\begin{aligned} f=\sum _{j=0}^{k-1}f_{j,k}. \end{aligned}$$
(1.2)

Moreover,

$$\begin{aligned} f_{j,k}\left( z\right) =\frac{1}{k}\sum _{l=0}^{k-1}\varepsilon ^{-jl}f\left( \varepsilon ^{l}z\right) ,\text { }z\in {\mathcal {G}}. \end{aligned}$$

By the uniqueness of this decomposition,  the functions \(f_{j,k} ,j=0,1,...,k-1,\) will be called \(\left( j,k\right) -\) symmetrical components of the function f.

In the next sections of the paper will be very useful the fact that \(f_{0,k}\in {\mathcal {H}}_{\mathcal{G}}(1)\) for \(f\in {\mathcal {H}}_{\mathcal{G}}(1).\) Note that

$$\begin{aligned} f_{0,k}(z)=1+\sum _{m=1}^{\infty }Q_{\left( f_{0,k}\right) ,m}(z)=1+\sum _{s=1}^{\infty }Q_{f,sk}(z),z\in {\mathcal {G}}. \end{aligned}$$

Note that the above unique decomposition (1.2) of functions was used in [20] to solve some functional equations, in [21] to construction a semi power series and in [22] to obtain a uniqueness theorem of Cartan type for holomorphic mappings in \({\mathbb {C}}^{n}.\)

We close this section with the following Golusin’s [10] result, very useful in the proof of the first result of Fekete–Szegö type for holomorphic functions of several complex variables.

Lemma 1.1

Let \(\Phi :U\rightarrow U\) be a holomorphic function of the form

$$\begin{aligned} \Phi \left( \zeta \right) =\sum _{\nu =0}^{\infty }a_{\nu }\zeta ^{\nu },\zeta \in U\text { .} \end{aligned}$$

Then

$$\begin{aligned} \left| a_{m}\right| \le 1-\left| a_{p}\right| ^{2}, \end{aligned}$$

for every \(m\in {\mathbb {N}}_{1},p\in {\mathbb {N}}_{0},\) satisfying the condition \(0\le 2p<m.\) The estimates are optimal.

Proof

Let us recall that the simplest case

$$\begin{aligned} \left| a_{1}\right| \le 1-\left| a_{0}\right| ^{2} \end{aligned}$$

follows from the well-known inequality

$$\begin{aligned} |\Phi ^{\prime }\left( \zeta \right) |\le \frac{1-|\Phi \left( \zeta \right) |^{2}}{1-|\zeta |^{2}},\zeta \in U.\text { } \end{aligned}$$

For another m, p we use a Krzyż’s idea [17, Chapt. 6.2] and some properties of (j, k)-symmetrical functions.

Let us take

$$\begin{aligned} \Psi \left( \zeta \right) =\zeta ^{m-2p}\Phi \left( \zeta \right) =\sum _{s=0}^{\infty }a_{s}\zeta ^{s+m-2p},\zeta \in U\text { }. \end{aligned}$$

Then \(\Psi :U\rightarrow U\) and is holomorphic. Thus the \((0,m-p)\)-part \(\Psi _{0,m-p}\) of \(\Psi \) transforms U into itself, is holomorphic and

$$\begin{aligned} \Psi _{0,m-p}(\zeta )=\sum _{s=0}^{\infty }a_{p+s\left( m-p\right) } \zeta ^{(s+1)\left( m-p\right) },\zeta \in U. \end{aligned}$$

Hence, by the Schwarz Lemma (in version with the zero \(\zeta =0\) of multiplicity \(m-p>0\)) it fulfils the inequality

$$\begin{aligned} \left| \Psi _{0,m-p}(\zeta )\right| \le \left| \zeta \right| ^{m-p},\zeta \in U. \end{aligned}$$

Therefore, the function

$$\begin{aligned} \Theta \left( \zeta \right) =\frac{1}{\zeta ^{m-p}}\Psi _{0,m-p}(\zeta ),\zeta \in U, \end{aligned}$$

maps U into itself, is holomorphic and has the expansion

$$\begin{aligned} \Theta \left( \zeta \right) =\sum _{s=0}^{\infty }a_{p+s\left( m-p\right) }\zeta ^{s\left( m-p\right) },\zeta \in U\text { }. \end{aligned}$$

Consequently, replacing \(\zeta ^{m-p}\in U\) by \(\xi \in \) U,  we get that the function

$$\begin{aligned} \sum _{s=0}^{\infty }b_{s}\xi ^{s}=\sum _{s=0}^{\infty }a_{p+s\left( m-p\right) }\xi ^{s},\xi \in U\text { } \end{aligned}$$

transforms holomorphically U into itself. Hence, by the first part of the proof, we get the thesis.

Note that the equality in the inequality is attained by the functions \(F(\zeta )=\) \(\zeta ^{m},\zeta \in U\) and \(F(\zeta )=\) \(\zeta ^{p},\zeta \in U.\) This completes the proof \(\square \)

2 Main results

We start this section with an n-dimensional Fekete–Szegö type theorem for bounded holomorphic functions on bounded complete n-circular domains in \({\mathbb {C}}^{n}.\) Note that it is a generalization of a 1-dimensional result given by Keogh and Merkes [14].

Theorem 2.1

Let \(\varphi \) be a function from the family

$$\begin{aligned} {\mathcal {B}}_{{\mathcal {G}}}\mathbb {(}0)=\left\{ \varphi \in {\mathcal {H}} _{{\mathcal {G}}}(0):|\varphi (z)|<1,z\in {\mathcal {G}}\right\} \end{aligned}$$

and has the form

$$\begin{aligned} \varphi (z)=\sum _{l=1}^{\infty }Q_{\varphi ,l}(z),z\in {\mathcal {G}}. \end{aligned}$$

Then, for every \(k\in {\mathbb {N}}_{1}\) and every \(\gamma \in {\mathbb {C}},\) there holds the inequality

$$\begin{aligned} \mu _{{\mathcal {G}}}\left( Q_{\varphi ,2k}-\gamma \left( Q_{\varphi ,k}\right) ^{2}\right) \le \max \{1,\left| \gamma \right| \}. \end{aligned}$$
(2.1)

The estimate is sharp.

Proof

For every arbitrarily fixed \(z\in {\mathcal {G}}\) we define the function

$$\begin{aligned} \Phi \left( \zeta \right) =\left\{ \begin{array} [l]{l} \frac{\varphi (\zeta z)}{\zeta },\zeta \in U\backslash \{0\}\\ Q_{\varphi ,1}(z)=\lim _{\zeta \rightarrow 0}\frac{\varphi (\zeta z)}{\zeta } ,\zeta =0 \end{array} \right. . \end{aligned}$$

Then \(\Phi \) is holomorphic,

$$\begin{aligned} \Phi \left( \zeta \right) =\sum _{\nu =0}^{\infty }a_{\nu }\zeta ^{\nu },\zeta \in U,\text { } \end{aligned}$$

\(a_{l-1}=Q_{\varphi ,l}(z),l\in {\mathbb {N}}_{1},\) and \(\left| \Phi \left( 0\right) \right| <1.\) Moreover, applying a \({\mathbb {C}}^{n}\)-version of Schwarz Lemma,  i.e., \(|\varphi (z)|\le \mu _{{\mathcal {G}}}(z),z\in {\mathcal {G}},\) for \(\varphi \in {\mathcal {B}}_{{\mathcal {G}}}\mathbb {(}0),\) we conclude also that for \(\zeta \in U\backslash \{0\}\)

$$\begin{aligned} \left| \Phi \left( \zeta \right) \right| =\frac{\left| \varphi (\zeta z)\right| }{\left| \zeta \right| }\le \frac{\mu _{{\mathcal {G}}}(\zeta z)}{\left| \zeta \right| }=\frac{\left| \zeta \right| \mu _{{\mathcal {G}}}(z)}{\left| \zeta \right| }<1, \end{aligned}$$

hence \(\Phi :U\longrightarrow U.\) Now let us observe that from the Lemma 1.1 we get

$$\begin{aligned} \left| a_{m-1}\right| \le 1-\left| a_{p-1}\right| ^{2}, \end{aligned}$$

for arbitrarily fixed \(m,p\in {\mathbb {N}}_{1},\) satisfying the condition \(0\le 2\left( p-1\right) <m-1.\) Thus

$$\begin{aligned} \left| Q_{\varphi ,m}(z)\right| \le 1-\left| Q_{\varphi ,p}(z)\right| ^{2},z\in {\mathcal {G}}. \end{aligned}$$

Therefore, for \(z\in {\mathcal {G}}\) and every \(\gamma \in {\mathbb {C}}\)

$$\begin{aligned}&\left| Q_{\varphi ,m}(z)-\gamma \left( Q_{\varphi ,p}(z)\right) ^{2}\right| \\&\quad \le \left| Q_{\varphi ,m}(z)\right| +\left| \gamma \right| \left| \left( Q_{\varphi ,p}(z)\right) ^{2}\right| \le 1-\left| Q_{\varphi ,p}(z)\right| ^{2}+\left| \gamma \right| \left| Q_{\varphi ,p}(z)\right| ^{2}\\&\quad =1+\left( \left| \gamma \right| -1\right) \left| Q_{\varphi ,p}(z)\right| ^{2}\le \max \{1,\left| \gamma \right| \}, \end{aligned}$$

because

$$\begin{aligned} \left\{ \begin{array} [l]{l} \left( \left| \gamma \right| -1\right) \left| Q_{\varphi ,p}(z)\right| ^{2}\le 0,\text { if \ }\left| \gamma \right| <1,\\ 0\le \left( \left| \gamma \right| -1\right) \left| Q_{\varphi ,p}(z)\right| ^{2}\le \left| \gamma \right| -1,\text { if }\left| \gamma \right| \ge 1 \end{array} \right. . \end{aligned}$$

Consequently, by the arbitrarinnes of \(z\in \mathcal {G}, \) also

$$\begin{aligned} \sup _{z\in {\mathcal {G}}}\left| Q_{\varphi ,m}(z)-\gamma \left( Q_{\varphi ,p}(z)\right) ^{2}\right| \le \max \{1,\left| \gamma \right| \}. \end{aligned}$$

Putting in the above \(p=k\) and \(m=2k,k\in {\mathbb {N}}_{1},\) we obtain \(m-1>2(p-1)\ge 1\ge 0\) and

$$\begin{aligned} \sup _{z\in {\mathcal {G}}}\left| Q_{\varphi ,2k}(z)-\gamma \left( Q_{\varphi ,k}(z)\right) ^{2}\right| \le \max \{1,\left| \gamma \right| \}. \end{aligned}$$

Finally, by the fact that \(Q_{\varphi ,2k}-\gamma \left( Q_{\varphi ,k}\right) ^{2}\) is a 2k-homogeneous polynomial for every \(\gamma \in {\mathbb {C}}\) and by the definition of its \(\mu _{{\mathcal {G}}}\)-balance, we get the statements of the Theorem 2.1.

Now, we will analyse the sharpness of the above estimate.

First, we prove that in the case \(\left| \gamma \right| \ge 1,\) the equality in (2.1) is attained by the function \(\varphi =\widetilde{\varphi }\in {\mathcal {B}}_{{\mathcal {G}}}\mathbb {(}0),{\widetilde{\varphi }}=I^{k},\) more precisely \({\widetilde{\varphi }}=I_{|{\mathcal {G}}}^{k}\) , i.e., \({\widetilde{\varphi }}(z)=I^{k}(z),z\in {\mathcal {G}}.\) Indeed, since \(Q_{{\widetilde{\varphi }},k}=I^{k},Q_{{\widetilde{\varphi }},2k}=0\) and \(\mu _{{\mathcal {G}}}(I^{2k})=1,\) we have

$$\begin{aligned} \mu _{{\mathcal {G}}}\left( Q_{\varphi ,2k}-\gamma \left( Q_{\varphi ,k}\right) ^{2}\right) =\mu _{{\mathcal {G}}}\left( -\gamma \left( Q_{\widetilde{\varphi },k}\right) ^{2}\right) =\left| \gamma \right| \mu _{{\mathcal {G}} }\left( \left( Q_{{\widetilde{\varphi }},k}\right) ^{2}\right) =\left| \gamma \right| =\max \{1,\left| \gamma \right| \}. \end{aligned}$$

Now, we show that in the case \(\left| \gamma \right| <1,\) the equality in (2.1) realizes the function \(\varphi ={\widehat{\varphi }} \in {\mathcal {B}}_{{\mathcal {G}}}\mathbb {(}0),{\widehat{\varphi }}=I^{2k},\) more precisely \({\widehat{\varphi }}=I_{|{\mathcal {G}}}^{2k}\) , i.e., \(\widehat{\varphi }(z)=I^{2k}(z),z\in {\mathcal {G}}.\) Indeed, since \(Q_{{\widehat{\varphi }} ,k}=0,Q_{{\widehat{\varphi }},2k}=I^{2k},\) we get

$$\begin{aligned} \mu _{{\mathcal {G}}}\left( Q_{\varphi ,2k}-\gamma \left( Q_{\varphi ,k}\right) ^{2}\right) =\mu _{{\mathcal {G}}}\left( Q_{{\widehat{\varphi }},2k}\right) =1=\max \{1,\left| \gamma \right| \}. \end{aligned}$$

This completes the proof. \(\square \)

Note that Theorem 2.1 generalizes an earlier result given by authors in [7].

In the sequel we apply Theorem 2.1 to study two Bavrin’s families \({\mathcal{M}}_{\mathcal{G}}, {\mathcal{N}}_{\mathcal{G}}\) of functions \(f\in {\mathcal {H}}_{{\mathcal {G}} }(1)\). These families are defined by the following family \({\mathcal {C}}_{\mathcal {G}},\)

$$\begin{aligned} {\mathcal {C}}_{\mathcal {G}}=\{f\in {\mathcal {H}}_{\mathcal {G}}(1):{\text {Re}}f(z)>0,z\in {\mathcal {G}}\} \end{aligned}$$

and by the following Temljakov [29] linear operator \({\mathcal {L}}\, :{\mathcal {H}}_{{\mathcal {G}}}\longrightarrow {\mathcal {H}}_{{\mathcal {G}}}\)

$$\begin{aligned} {\mathcal {L}}\,f(z)=f(z)+Df(z)(z),z\in {\mathcal {G}}, \end{aligned}$$

where Df(z) means the Fréchet derivative of f at the point z. Note that the operator \({\mathcal {L}}\,\) is invertible and

$$\begin{aligned} {\mathcal {L}}\,^{-1}f(z)=\int \limits _{0}^{1}f(zt)\mathrm{{d}}t,z\in {\mathcal {G}}. \end{aligned}$$

It is obvious also that for the transforms \({\mathcal {L}}\, f,{{\mathcal {L}}\,}{{\mathcal {L}}\,}f\) of the functions \(f\in {\mathcal{H}}_{\mathcal{G}}(1)\) we have

$$\begin{aligned}&{\mathcal {L}}\,f(z)=1+\sum _{m=1}^{\infty }Q_{{\mathcal {L}}\,f,m}(z)=1+\sum _{m=1}^{\infty }(m+1)Q_{f,m}(z),z\in {\mathcal {G}}, \end{aligned}$$
(2.2)
$$\begin{aligned}&\quad {{\mathcal {L}}\,}{{\mathcal {L}}\,}f(z)=1+\sum _{m=1}^{\infty }Q_{{{\mathcal {L}}\,}{{\mathcal {L}}\,}f,m}(z)=1+\sum _{m=1}^{\infty }(m+1)^{2}Q_{f,m}(z),z\in {\mathcal {G}}. \end{aligned}$$
(2.3)

Moreover,

$$\begin{aligned} {\mathcal {L}}\,^{-1}f(z)=1+\sum _{m=1}^{\infty }Q_{{\mathcal {L}}\,^{-1}f,m} (z)=1+\sum _{m=1}^{\infty }\frac{1}{m+1}Q_{f,m}(z), z\in {\mathcal {G}}. \end{aligned}$$
(2.4)

We say that a function \(f\in {\mathcal {H}}_{{\mathcal {G}}}(1)\) belongs to the Bavrin’s family \({\mathcal{M}}_{{\mathcal{G}}}\) \(({\mathcal{N}}_{{\mathcal{G}}})\) if it satisfies the factorization

$$\begin{aligned}&{\mathcal {L}}\,f(z)=f(z)h(z),\text { }z\in {\mathcal {G}}, \nonumber \\& {{\mathcal {L}}\,}{{\mathcal {L}}\,}f(z)={\mathcal {L}}\,f(z)h(z),\text { }z\in {\mathcal {G}}, \end{aligned}$$
(2.5)

together with a function \(h\in {\mathcal{C}}_{{\mathcal{G}}}\) and the transform \({\mathcal {L}}\,f,\) \(({{\mathcal {L}}\,}{{\mathcal {L}}\,}f),\) respectively. Note that the families \({\mathcal{M}}_{{\mathcal{G}}},\) \({\mathcal{N}}_{{\mathcal{G}}}\) correspond with the well-known families of normalized univalent starlike (convex) functions in the disc U [2] and the family \({\mathcal{M}}_{{\mathcal{G}}}\) can be used to construction biholomorphic starlike mappings in \({\mathbb {C}}^{n}\) (see [6, 18], compare also [12, 25]). Between functions from \({\mathcal {M}}_{{\mathcal {G}}},{\mathcal {N}}_{{\mathcal {G}}}\) there holds a relationship, corresponding to the well-known Alexander type connexion [1], for univalent starlike and convex mappings in the unit disc. Here, this relationship is the following: if \(f\in {\mathcal {N}}_{{\mathcal {G}}},\) then \({\mathcal {L}}\,f\in {\mathcal {M}}_{{\mathcal {G}}}\) and conversely, if \(f\in {\mathcal {M}}_{{\mathcal {G}}},\) then \({\mathcal {L}}\,^{-1}f\in {\mathcal {M}} _{{\mathcal {G}}}\) [2].

We begin the presentation of some Fekete–Szegö type results in Bavrin’s families with the following theorem.

Theorem 2.2

Let \({\mathcal{G}} \subset {\mathbb{C}}^{n} \) be a bounded complete n-circular domain and let \(f\in {\mathcal {N}}_{{\mathcal {G}}}\cap {\mathcal {F}}_{0,k} \mathbf {(}{\mathcal {G}}\mathbf {,}\mathbb {C)},\) \(k\in {\mathbb {N}}_{2}.\) If the expansion of the function f into a series of m-homogenous polynomials \(Q_{f,m}\) has the form (1.1), with \(Q_{f,0}=1,\) then for the homogeneous polynomials \(Q_{f,2k},Q_{f,k}\) and every \(\lambda \in {\mathbb {C}}\) there holds the following sharp estimate:

$$\begin{aligned} \mu _{{\mathcal {G}}}\left( Q_{f,2k\;}-\lambda \left( Q_{f,k}\right) ^{2}\right) \le \frac{1}{k\left( 2k+1\right) }\max \left\{ 1,\left| \frac{4\left( 2k+1\right) \lambda -\left( k+1\right) ^{2}\left( k+2\right) }{k\left( k+1\right) ^{2}}\right| \right\} . \end{aligned}$$
(2.6)

Proof

Let us recall that between the functions \(p\in \) \({\mathcal{C}}_{{\mathcal{G}}}\) and \(\varphi \in {\mathcal {B}}_{{\mathcal {G}}}\mathbb {(}0),\) there holds the following relation [2]:

$$\frac{p-1}{p+1}=\varphi \in {\mathcal{B}}_{{\mathcal {G}}}(0)\Longleftrightarrow p\in {\mathcal{C}}_{{\mathcal{G}}}. $$

Let \(k\in {\mathbb {N}}_{2}\) be arbitrarily fixed and let the function f belongs to \({\mathcal {N}}_{{\mathcal {G}}}\cap {\mathcal {F}}_{0,k}\mathbf {(}{\mathcal {G}} \mathbf {,}\mathbb {C)}.\) Then, by the definition of the family \({\mathcal {N}} _{{\mathcal {G}}}\) and by the above relation between the families \( {\mathcal{C}}_{\mathcal{G}},{\mathcal {B}}_{{\mathcal {G}}}\mathbb {(}0),\) we get

$$\begin{aligned} \frac{{{\mathcal {L}}\,}{{\mathcal {L}}\,}f(z)-{\mathcal {L}}\,f(z)}{{{\mathcal {L}}\,}{{\mathcal {L}}\,}f(z)+{\mathcal {L}}\, f(z)}=\varphi \left( z\right) ,z\in {\mathcal {G}}. \end{aligned}$$
(2.7)

where \(\varphi \in {\mathcal {B}}_{{\mathcal {G}}}\mathbb {(}0) {\cap {\mathcal{F}}} _{0,k}\mathbf {(}{\mathcal {G}}\mathbf {,}{\mathbb {C}}).\) On the other hand, from (1.1) we have for \(z\in {\mathcal {G}}\):

$$\begin{aligned} \varphi \left( z\right)&=\sum _{s=1}^{\infty }Q_{\varphi ,sk}(z),\\ f\left( z\right)&=1+\sum _{s=1}^{\infty }Q_{f,sk}(z) \end{aligned}$$

and from (2.2) , (2.3) also

$$\begin{aligned} {\mathcal {L}}\,f(z)&=1+\sum _{s=1}^{\infty }\left( sk+1\right) Q_{f,sk}(z),\\ {{\mathcal {L}}\,}{{\mathcal {L}}\,}f(z)&=1+\sum _{s=1}^{\infty }\left( sk+1\right) ^{2} Q_{f,sk}(z). \end{aligned}$$

Inserting the above expansion of functions into (2.7) , we receive after computations

$$\begin{aligned} \sum _{s=1}^{\infty }sk\left( 1+sk\right) Q_{f,sk}(z)=\left( \sum _{s=1}^{\infty }Q_{\varphi ,sk}(z)\right) \left( 2+\sum _{s=1}^{\infty }\left( sk+1\right) \left( sk+2\right) Q_{f,sk}(z)\right) . \end{aligned}$$

Then, comparing the m-homogeneous polynomials of the same degree on both sides of the above equality, we can determine homogeneous polynomials \(Q_{\varphi ,k},Q_{\varphi ,2k},\) as follows

$$\begin{aligned} Q_{\varphi ,k}&=\frac{1}{2}k\left( k+1\right) Q_{f,k}, \nonumber \\ Q_{\varphi ,2k}&=k\left( 2k+1\right) Q_{f,2k}-\frac{1}{4}k\left( k+1\right) ^{2}\left( k+2\right) \left( Q_{f,k}\right) ^{2}. \end{aligned}$$
(2.8)

Putting the above equalities into Theorem 2.1 and using the fact that the mapping \(\left( Q_{f,k}\right) ^{2},\) is a 2k-homogenous polynomial, we obtain

$$\begin{aligned} \mu _{{\mathcal {G}}}\left[ k\left( 2k+1\right) Q_{f,2k}-\left( \frac{1}{4}k\left( k+1\right) ^{2}\left( k+2\right) +\gamma \frac{1}{4}k^{2}\left( k+1\right) ^{2}\right) \left( Q_{f,k}\right) ^{2}\right] \le \max \{1,\left| \gamma \right| \}. \end{aligned}$$

Denoting

$$\begin{aligned} \lambda =\frac{\left( k+1\right) ^{2}}{4\left( 2k+1\right) }\left( k+2+\gamma k\right) , \end{aligned}$$

we obtain

$$\begin{aligned} \mu _{{\mathcal {G}}}\left( Q_{f,2k}-\lambda \left( Q_{f,k}\right) ^{2}\right) \le \frac{1}{k\left( 2k+1\right) }\max \left\{ 1,\left| \frac{4\left( 2k+1\right) \lambda -\left( k+1\right) ^{2}\left( k+2\right) }{k\left( k+1\right) ^{2}}\right| \right\} . \end{aligned}$$

Now, we show the sharpness of our estimate. To do it, let us consider two cases.

At the beginning, we prove that, in the case

$$\begin{aligned} \frac{\left| 4\left( 2k+1\right) \lambda -\left( k+1\right) ^{2}\left( k+2\right) \right| }{k\left( k+1\right) ^{2}}\ge 1 \end{aligned}$$

the equality in (2.6) is attained by the function \(f={\mathcal {L}}\, ^{-1}{\widetilde{f}},\) with

$$\begin{aligned} {\widetilde{f}}(z)=\left( 1-I^{k}(z)\right) ^{\frac{-2}{k}},z\in {\mathcal {G}}, \end{aligned}$$
(2.9)

where the branch of the function \(\left( 1-\xi \right) ^{\frac{-2}{k}}\) takes value 1 at the point \(\xi =0.\) Indeed. Function f belongs to \({\mathcal {F}} _{0,k}\mathbf {(}{\mathcal {G}}\mathbf {,}\mathbb {C)},\) because \({\widetilde{f}} \in {\mathcal {F}}_{0,k}\mathbf {(}{\mathcal {G}}\mathbf {,}\mathbb {C)}.\) On the other hand, the function \(f={\mathcal {L}}\,^{-1}{\widetilde{f}}\) belongs to \({\mathcal {N}} _{{\mathcal {G}}},\) because the function \(\varphi \) from (2.7) has the form \(\varphi ={\widetilde{\varphi }}=I^{k}\) and belongs to \({\mathcal {B}} _{{\mathcal {G}}}\mathbb {(}0).\) Therefore, we can write equalities (2.8 ) in the following form

$$\begin{aligned} Q_{f,k}=\frac{2}{k(k+1)}I^{k},Q_{f,2k}=\frac{k+2}{k^{2}(2k+1)}\left( I^{k}\right) ^{2}. \end{aligned}$$
(2.10)

From this, by the case condition for \(\lambda ,\) we have step by step:

$$\begin{aligned} \mu _{_{{\mathcal {G}}}}(Q_{f,2k}-\lambda \left( Q_{f,k}\right) ^{2})&=\mu _{_{{\mathcal {G}}}}\left[ \frac{\left( k+1\right) ^{2}\left( k+2\right) -4\left( 2k+1\right) \lambda }{k^{2}\left( k+1\right) ^{2}(2k+1)}\left( I^{k}\right) ^{2}\right] \\&=\left| \frac{\left( k+1\right) ^{2}\left( k+2\right) -4\left( 2k+1\right) \lambda }{k^{2}\left( k+1\right) ^{2}(2k+1)}\right| \mu _{_{{\mathcal {G}}}}\left( I^{2k}\right) \\&=\frac{1}{k(2k+1)}\max \left\{ 1,\left| \frac{4\left( 2k+1\right) \lambda -\left( k+1\right) ^{2}\left( k+2\right) }{k\left( k+1\right) ^{2}}\right| \right\} . \end{aligned}$$

Now, we show that, in the case

$$\begin{aligned} \frac{\left| 4\left( 2k+1\right) \lambda -\left( k+1\right) ^{2}\left( k+2\right) \right| }{k\left( k+1\right) ^{2}}<1 \end{aligned}$$

the equality in (2.6) realizes the function \(f={\mathcal {L}}\, ^{-1}{\widehat{f}},\) with

$$\begin{aligned} {\widehat{f}}(z)=\left( 1-I^{2k}(z)\right) ^{\frac{-1}{k}},z\in {\mathcal {G}}, \end{aligned}$$
(2.11)

where the branch of the function \(\left( 1-\xi \right) ^{\frac{-1}{k}}\) takes value 1 at the point \(\xi =0.\) The function \(f={\mathcal {L}}\,^{-1}{\widehat{f}}\) belongs to \({\mathcal {F}}_{0,k}\mathbf {(}{\mathcal {G}}\mathbf {,}\mathbb {C)},\) because \({\widehat{f}}\in {\mathcal {F}}_{0,k}\mathbf {(}{\mathcal {G}}\mathbf {,} \mathbb {C)}.\) Also \(f\in \) \({\mathcal {N}}_{{\mathcal {G}}},\) because the function \(\varphi \) from (2.7) has the form \(\varphi ={\widehat{\varphi }}=I^{2k}\) and belongs to \({\mathcal {B}}_{{\mathcal {G}}}\mathbb {(}0).\) Therefore, we can write equalities (2.8) in the following form

$$\begin{aligned} Q_{f,k}=0,Q_{f,2k}=\frac{1}{k(2k+1)}I^{2k}. \end{aligned}$$
(2.12)

From this, by the case condition for \(\lambda ,\) we have:

$$\begin{aligned} \mu _{{\mathcal {G}}}(Q_{f,2k}-\lambda \left( Q_{f,k}\right) ^{2})&=\mu _{{\mathcal {G}}}\left( \frac{1}{k(2k+1)}I^{2k}\right) =\frac{1}{k(2k+1)}\mu _{_{{\mathcal {G}}}}\left( I^{2k}\right) \\&=\frac{1}{k(2k+1)}\max of\left\{ 1,\left| \frac{4\left( 2k+1\right) \lambda -\left( k+1\right) ^{2}\left( k+2\right) }{k\left( k+1\right) ^{2}}\right| \right\} . \end{aligned}$$

This completes the proof. \(\square \)

We continue the presentation of some Fekete–Szegö type results in Bavrin’s families with the following theorem:

Theorem 2.3

Let \( {{\mathcal{G}}\subset }\) \({\mathbb {C}}^{n}\) be a bounded complete n-circular domain and let \(k\in {\mathbb {N}}_{2}.\) If the expansion of the function \(f\in {\mathcal {M}}_{{\mathcal {G}}}\cap {\mathcal {F}}_{0,k}\mathbf {(} {\mathcal {G}}\mathbf {,}\mathbb {C)}\), into a series of m-homogenous polynomials \(Q_{f,m}\) has the form (1.1), with \(Q_{f,0}=1,\) then for the homogeneous polynomials \(Q_{f,2k},Q_{f,k}\) and \(\lambda \in {\mathbb {C}}\) there holds the following sharp estimate:

$$\begin{aligned} \mu _{{\mathcal {G}}}\left( Q_{f,2k\;}-\lambda \left( Q_{f,k}\right) ^{2}\right) \le \frac{1}{k}\max \left\{ 1,\frac{\left| 2+k-4\lambda \right| }{k}\right\} . \end{aligned}$$
(2.13)

Proof

Let \(k\in {\mathbb {N}}_{2}\) be arbitrarily fixed. Then, it is obvious that \({\mathcal {L}}\,^{-1}f\) \(\in {\mathcal {F}}_{0,k}\mathbf {(}{\mathcal {G}}\mathbf {,} \mathbb {C)},\) because \(f\in {\mathcal {F}}_{0,k}\mathbf {(}{\mathcal {G}} \mathbf {,}\mathbb {C)}.\) Also the assumption that f \(\in {\mathcal {M}} _{{\mathcal {G}}},\) by the relationship of the Alexander type, gives that \({\mathcal {L}}\,^{-1}f\) \(\in {\mathcal {N}}_{{\mathcal {G}}}.\) Hence, we have that \({\mathcal {L}}\,^{-1}f\) \(\in {\mathcal {N}}_{{\mathcal {G}}}\cap {\mathcal {F}} _{0,k}\mathbf {(}{\mathcal {G}}\mathbf {,}\mathbb {C)}.\) On the other hand, its expansion into a series of m-homogenous polynomials \(Q_{f,m},\) by ( 2.4),  has the form

$$\begin{aligned} {\mathcal {L}}\,^{-1}f(z)=1+\sum _{s=1}^{\infty }\frac{1}{sk+1}Q_{f,sk} (z),z\in {\mathcal {G}}. \end{aligned}$$

Thus, in view of (2.6) from Theorem 2.2, we get that for every \(\delta \in {\mathbb {C}}\)

$$\begin{aligned}&\mu _{{\mathcal {G}}}\left( \frac{1}{2k+1}Q_{f,2k\;}-\delta \left( \frac{1}{k+1}Q_{f,k}\right) ^{2}\right) \\&\quad \le \frac{1}{k\left( 2k+1\right) } \max \left\{ 1,\left| \frac{4\left( 2k+1\right) \delta -\left( k+1\right) ^{2}\left( k+2\right) }{k\left( k+1\right) ^{2}}\right| \right\} . \end{aligned}$$

Hence,

$$\begin{aligned} \mu _{{\mathcal {G}}}\left( Q_{f,2k\;}-\delta \frac{2k+1}{\left( k+1\right) ^{2}}\left( Q_{f,k}\right) ^{2}\right) \le \frac{1}{k}\max \left\{ 1,\left| \frac{4\left( 2k+1\right) \delta -\left( k+1\right) ^{2}\left( k+2\right) }{k\left( k+1\right) ^{2}}\right| \right\} \end{aligned}$$

and denoting \(\delta \frac{2k+1}{\left( k+1\right) ^{2}}=\) \(\lambda ,\) finally

$$\begin{aligned} \mu _{{\mathcal {G}}}\left( Q_{f,2k\;}-\lambda \left( Q_{f,k}\right) ^{2}\right) \le \frac{1}{k}\max \left\{ 1,\left| \frac{4\lambda -\left( k+2\right) }{k}\right| \right\} , \end{aligned}$$

which is the same as (2.13).

It remains to show the sharpness of the estimate (2.13).

First, we prove that, in the case

$$\begin{aligned} \frac{\left| 2+k-4\lambda \right| }{k}\ge 1 \end{aligned}$$

the equality in (2.13) is attained by the function \(f={\widetilde{f}}\) defined in (2.9). Of course \({\widetilde{f}}\in {\mathcal {F}} _{0,k}\mathbf {(}{\mathcal {G}}\mathbf {,}\mathbb {C)}\) and, by the Alexander type relationship and the fact that \({\mathcal {L}}\,^{-1}{\widetilde{f}}\in \) \({\mathcal {N}}_{{\mathcal {G}}}\) (see the proof of the sharpness in Theorem 2.2),  the function \({\widetilde{f}}\) belongs to \({\mathcal {M}}_{{\mathcal {G}}}.\) Therefore, in view of (2.2) and (2.10),  we achieve

$$\begin{aligned} Q_{f,k}=Q_{{\widetilde{f}},k}=\frac{2}{k}I^{k},\text { }Q_{f,2k}=Q_{\widetilde{f},2k}=\frac{k+2}{k^{2}}\left( I^{k}\right) ^{2}. \end{aligned}$$

From this (see the proof of the sharpness in Theorem 2.2) and by the case condition for \(\lambda ,\) we have step by step:

$$\begin{aligned} \mu _{_{{\mathcal {G}}}}(Q_{f,2k}-\lambda \left( Q_{f,k}\right) ^{2})&=\mu _{_{{\mathcal {G}}}}\left[ \frac{\left( k+2\right) -4\lambda }{k^{2} }\left( I^{k}\right) ^{2}\right] \\&=\left| \frac{\left( k+2\right) -4\lambda }{k^{2}}\right| \mu _{_{{\mathcal {G}}}}\left( I^{2k}\right) =\frac{1}{k}\max \left\{ 1,\frac{\left| 2+k-4\lambda \right| }{k}\right\} . \end{aligned}$$

Now, we show that, in the case

$$\begin{aligned} \frac{\left| 2+k-4\lambda \right| }{k}<1 \end{aligned}$$

the equality in (2.13) realizes the function \(f={\widehat{f}}\) defined in (2.11). Of course \({\widehat{f}}\in {\mathcal {F}}_{0,k}\mathbf {(} {\mathcal {G}}\mathbf {,}\mathbb {C)}\) and, by the Alexander type relationship and the fact that \({\mathcal {L}}\,^{-1}{\widehat{f}}\in \) \({\mathcal {N}}_{{\mathcal {G}}}\) (see the proof of the sharpness in Theorem 2.2),  the function \(\widehat{f}\) belongs to \({\mathcal {M}}_{{\mathcal {G}}}.\) Therefore, in view of ( 2.2) and (2.12) we achieve

$$\begin{aligned} Q_{f,k}=Q_{{\widehat{f}},k}=0,\text { }Q_{f,2k}=Q_{{\widehat{f}},2k}=\frac{1}{k}I^{2k}. \end{aligned}$$

From this (see the proof of the sharpness in Theorem 2.2) and by the case condition for \(\lambda ,\) we conclude that:

$$\begin{aligned} \mu _{_{{\mathcal {G}}}}(Q_{f,2k}-\lambda \left( Q_{f,k}\right) ^{2} )=\mu _{_{{\mathcal {G}}}}\left( \frac{1}{k}I^{2k}\right) =\frac{1}{k}=\frac{1}{k}\max \left\{ 1,\frac{\left| 2+k-4\lambda \right| }{k}\right\} . \end{aligned}$$

This completes the proof. \(\square \)

Now, we transfer the statement of Theorem 2.3, onto a family \({\mathcal {M}} _{{\mathcal {G}}}^{k}\cap {\mathcal {F}}_{0,k}\mathbf {(}{\mathcal {G}}\mathbf {,} \mathbb {C)}\mathbf {,}k\in {\mathbb {N}}_{2}.\) Here, \({\mathcal {M}}_{{\mathcal {G}}} ^{k}\) is defined by the factorization similar as for the elements from \({\mathcal {M}}_{{\mathcal {G}}}.\) More precisely, the function f of the right hand side in (2.5) is replaced by a function from \({\mathcal {F}} _{0,k}\mathbf {(}{\mathcal {G}}\mathbf {,}\mathbb {C)},\) generated by f. Formally, we say that a function \(f\in {\mathcal {H}}_{{\mathcal {G}}}(1)\) belongs to \({\mathcal {M}}_{{\mathcal {G}}}^{k},k\in {\mathbb {N}}_{2},\) (see [3, 5]) if there exists a function \(h\in {\mathcal{C}}_{\mathcal{G}}\) such that

$$\begin{aligned} {\mathcal {L}}\,f(z)=f_{0,k}(z)h(z),z\in {\mathcal {G}}, \end{aligned}$$

where \(f_{0,k}\in {\mathcal {F}}_{0,k}\mathbf {(}{\mathcal {G}}\mathbf {,}\mathbb {C)}\) is the \(\left( 0,k\right) \)- symmetrical component of the function f in the decomposition (1.2) from Theorem A. This family for \(k=2\) corresponds to a well-known Sakaguchi family [27] of a complex variable functions; strictly speaking of functions univalent starlike with respect to two symmetric points. In the paper [5] it was shown that for \(k\in {\mathbb {N}}_{2}\) the inclusions \({\mathcal{M}}_{\mathcal{G}}\subset {\mathcal {M}} _{{\mathcal {G}}}^{k},\) \({\mathcal {M}}_{{\mathcal {G}}}^{k}\subset {\mathcal{M}}_{\mathcal{G}}\) do not hold, but

$$\begin{aligned} {\mathcal{M}}_{\mathcal{G}}\cap {\mathcal {F}}_{0,k}\mathbf {(}{\mathcal {G}}\mathbf {,} \mathbb {C)}={\mathcal {M}}_{{\mathcal {G}}}^{k}\cap {\mathcal {F}}_{0,k}\mathbf {(} {\mathcal {G}}\mathbf {,}\mathbb {C)}. \end{aligned}$$

This identity and Theorem 2.3 implies directly the next result of Fekete–Szegö type in Bavrin’s families:

Theorem 2.4

Let \( {{\mathcal{G}}\subset }\) \({\mathbb {C}}^{n}\) be a bounded complete n-circular domain and let \(f\in {\mathcal {M}}_{{\mathcal {G}}}^{k}\cap {\mathcal {F}} _{0,k}\mathbf {(}{\mathcal {G}}\mathbf {,}\mathbb {C)},k\in {\mathbb {N}}_{2} .\) If the expansion of the function f into a series of m-homogenous polynomials \(Q_{f,m}\) has the form (1.1), with \(Q_{f,0}=1,\) then for the homogeneous polynomials \(Q_{f,2k},Q_{f,k}\) and \(\lambda \in {\mathbb {C}}\) there holds the following sharp estimate:

$$\begin{aligned} \mu _{{\mathcal {G}}}\left( Q_{f,2k\;}-\lambda \left( Q_{f,k}\right) ^{2}\right) \le \frac{1}{k}\max \left\{ 1,\frac{\left| 2+k-4\lambda \right| }{k}\right\} . \end{aligned}$$

The equality in the above inequality realize the same functions \(f=\widetilde{f},f={\widehat{f}}\)as in the previous Theorem 2.2.

3 Applications

In this section we apply Theorems 2.3 and 2.4, to obtain a Fekete–Szegö type results for two families of biholomorphic mappings in \({\mathbb {C}}^{n}.\) By \(S^{*}({\mathbb {B}}^{n})\) let us denote the family of biholomorphic mappings \(F\in {\mathcal {F}}({\mathbb {B}}^{n},{\mathbb {C}}^{n}),\) \(F\left( 0\right) =0,\) \(DF\left( 0\right) =I\) onto starlike domains \(F\left( {\mathbb {B}}^{n}\right) .\) For a wide collection of references in this area see the monographs [11, 16]. In a Kikuchi–Matsuno–Suffridge characterization [15, 23, 28] of the family \(S^{*}({\mathbb {B}}^{n})\), the collection \(P\left( {\mathbb {B}}^{n}\right) \) of all holomorphic mappings \(H\in {\mathcal {F}} ({\mathbb {B}}^{n},{\mathbb {C}}^{n}),H(0)=0,DH(0)=I,\) such that \({\text {}}{Re} \left\langle H\left( z\right) ,z\right\rangle >0,z\in {\mathbb {B}} ^{n}\backslash \left\{ 0\right\} ,\) plays the main role (here \(\left\langle \cdot ,\cdot \right\rangle \) means the Euclidean inner product). This characterization is included in the following theorem:

Theorem B

A locally biholomorphic mapping \(F\in {\mathcal {F}}({\mathbb {B}} ^{n},{\mathbb {C}}^{n})\) normalized by the conditions \(F(0)=0,DF(0)=I,\) belongs to \(S^{*}({\mathbb {B}}^{n}),\) iff there exist a mapping \(H\in P\left( {\mathbb {B}}^{n}\right) \) such that

$$\begin{aligned} F\left( z\right) =DF\left( z\right) H\left( z\right) ,z\in {\mathbb {B}} ^{n}. \end{aligned}$$
(3.1)

Let \(\widetilde{S^{*}}({\mathbb {B}}^{n})\) be the family of mappings \(F\in S^{*}({\mathbb {B}}^{n})\mathbb {\ }\) with the factorization

$$\begin{aligned} F\left( z\right) =z f(z),z\in {\mathbb {B}}^{n}, \end{aligned}$$
(3.2)

where \(f\in {\mathcal {H}}_{{\mathbb {B}}^{n}}(1)\).

In the paper [6] the authors considered a family of biholomorphic mappings \(S^{k}({\mathbb {B}}^{n}),k\in {\mathbb {N}}_{2},\) defined by an equation similar to (3.1). More precisely, the mapping F of the left hand side in (3.1) is replaced by a function from \({\mathcal {F}} _{1,k}\left( {\mathbb {B}}^{n},{\mathbb {C}}^{n}\right) ,\) generated by F.

Formally, we say that a locally biholomorphic mapping \(F\in {\mathcal {F}} ({\mathbb {B}}^{n},{\mathbb {C}}^{n}),\) normalized by \(F(0)=0,DF(0)=I,\) belongs to the family \(S^{k}({\mathbb {B}}^{n}),k\in {\mathbb {N}}_{2},\) if it satisfies the equation

$$\begin{aligned} F_{1,k}(z)=DF(z)H(z),\text { }z\in {\mathbb {B}}^{n}, \end{aligned}$$

where \(H\in P\left( {\mathbb {B}}^{n}\right) \) and \(F_{1,k}\in {\mathcal {F}} _{1,k}({\mathbb {B}}^{n},{\mathbb {C}}^{n})\) is the \(\left( 1,k\right) \)-symmetrical part of F in the decomposition (1.2). Also in the paper [6] the authors proved, that for every \(k\in {\mathbb {N}}_{2}\) any inclusions \(S^{k}({\mathbb {B}}^{n})\subset S^{*}({\mathbb {B}}^{n}),\) \(S^{*}({\mathbb {B}}^{n})\subset S^{k}({\mathbb {B}}^{n})\) do not holds. However, for the same k there holds the following identity:

$$\begin{aligned} S^{k}({\mathbb {B}}^{n})\cap {\mathcal {F}}_{1,k}\left( {\mathbb {B}}^{n} ,{\mathbb {C}}^{n}\right) =S^{*}({\mathbb {B}}^{n})\cap {\mathcal {F}}_{1,k}\left( {\mathbb {B}}^{n},{\mathbb {C}}^{n}\right) . \end{aligned}$$
(3.3)

Let \(\widetilde{S^{k}}({\mathbb {B}}^{n}),k\in {\mathbb {N}}_{2},\) be the family of mappings \(F\in S^{k}({\mathbb {B}}^{n})\mathbb {\ }\) with the factorization (3.2). Now, we present the main theorem, in this section of the paper. It is a Fekete–Szegö type result for locally biholomorphic mappings in \({\mathbb {C}}^{n},\) compare [30].

Theorem 3.1

For mappings \(F\in \widetilde{S^{k}}({\mathbb {B}}^{n})\) \(\cap {\mathcal {F}} _{1,k}\left( {\mathbb {B}}^{n},{\mathbb {C}}^{n}\right) ,k\in {\mathbb {N}}_{2},\) the parameter \(\lambda \in {\mathbb {C}}\) and points \(z\in {\mathbb {B}}^{n}\backslash \left\{ 0\right\} \) there holds the following sharp estimate

$$\begin{aligned} \left| \frac{T_{z}\left( D^{2k+1}F(0)(z^{2k+1})\right) }{\left( 2k+1\right) !\left\| z\right\| ^{2k+1}}-\lambda \left( \frac{T_{z}\left( D^{k+1}F(0)(z^{k+1})\right) }{\left( k+1\right) !\left\| z\right\| ^{k+1}}\right) ^{2}\right| \le \frac{1}{k}\max \left\{ 1,\frac{\left| 2+k-4\lambda \right| }{k}\right\} , \end{aligned}$$
(3.4)

where \(T_{z}\in \left( {\mathbb {C}}^{n}\right) ^{*},z\in {\mathbb {B}} ^{n}\backslash \left\{ 0\right\} ,\) is arbitrary functional satisfying the conditions \(\left\| T_{z}\right\| =1,\) \(T_{z}\left( z\right) =\left\| z\right\| .\)

Proof

We start with a few facts, very useful in the proof:

1. A mapping F,  satisfying (3.2) , belongs to \(S^{k}({\mathbb {B}}^{n}),\) iff \(f\in {\mathcal {M}}_{{\mathbb {B}}^{n}}^{k}\) (see [6, 18]).

2. A mapping F of the form (3.2) belongs to \({\mathcal {F}}_{1,k}\left( {\mathbb {B}}^{n},{\mathbb {C}}^{n}\right) ,\) iff \(f\in \) \({\mathcal {F}}_{0,k}\left( {\mathbb {B}}^{n},{\mathbb {C}}\right) .\)

3. For \(z\in {\mathbb {B}}^{n}\) there hold (see for instance [30]) the equalities:

$$\begin{aligned}&\frac{D^{2k+1}F(0)(z^{2k+1})}{\left( 2k+1\right) !}=z\frac{D^{2k} f(0)(z^{2k})}{\left( 2k\right) !},\\&\quad \frac{D^{k+1}F(0)(z^{k+1})}{\left( k+1\right) !}=z\frac{D^{k}f(0)(z^{k} )}{\left( k\right) !}. \end{aligned}$$

4. For \(\mathbb {\lambda \in {\mathbb {C}}}\) the mapping \(Q_{f,2k}-\lambda Q_{f,k}^{2}\in {\mathcal {F}}({\mathbb {B}}^{n},{\mathbb {C}}),\) is a 2kth homogeneous polynomial.

5. There hold the identity \(\mu _{{\mathbb {B}}^{n}}(\cdot )=||\cdot ||\) in \({\mathbb {C}}^{n}\).

Applying the above facts and Theorem 2.5, we get step by step

$$\begin{aligned}&\left| \frac{T_{z}\left( D^{2k+1}F(0)(z^{2k+1})\right) }{\left( 2k+1\right) !\left\| z\right\| ^{2k+1}}-\lambda \left( \frac{T_{z}\left( D^{k+1}F(0)(z^{k+1})\right) }{\left( k+1\right) !\left\| z\right\| ^{k+1}}\right) ^{2}\right| \\&\quad =\left| \frac{T_{z}\left( z\right) D^{2k}f(0)(z^{2k})}{\left( 2k\right) !\left\| z\right\| ^{2k+1}}-\lambda \left( \frac{T_{z}\left( z\right) D^{k}f(0)(z^{k})}{\left( k\right) !\left\| z\right\| ^{k+1} }\right) ^{2}\right| \\&\quad =\left| \frac{D^{2k}f(0)(z^{2k})}{\left( 2k\right) !\left\| z\right\| ^{2k}}-\lambda \left( \frac{D^{k} f(0)(z^{k})}{\left( k\right) !\left\| z\right\| ^{k}}\right) ^{2}\right| \\&\quad =\left| \frac{Q_{f,2k}(z)}{\left\| z\right\| ^{2k}} -\lambda \left( \frac{Q_{f,k}(z)}{\left\| z\right\| ^{k}}\right) ^{2}\right| \\&\quad \le \sup _{z\in {\mathbb {B}}^{n}}\frac{\left| Q_{f,2k} (z)-\lambda \left( Q_{f,k}(z)\right) ^{2}\right| }{\left\| z\right\| ^{2k}}=\mu _{{\mathbb {B}}^{n}}\left( Q_{f,2k}-\lambda \left( Q_{f,k}\right) ^{2}\right) \\&\quad \le \frac{1}{k}\max \left\{ 1,\frac{\left| 2+k-4\lambda \right| }{k}\right\} . \end{aligned}$$

It remains to show the sharpness of the estimate (3.4).

To do it, observe first that there exist points \(z^{0}\in {\mathbb {B}} ^{n}\backslash \left\{ 0\right\} \) such that \(|I(\frac{z^{0}}{\left\| z^{0}\right\| })|=1.\) It follows from the maximum principle for the modulus of holomorphic functions of several complex variables,  because \(\left\| I\right\| =\mu _{{\mathbb {B}}^{n}}(I)=1.\) We will show that the equality in (3.4) in such points are attained by the mappings \(\widetilde{F},{\widehat{F}}\) of the form (3.2), where \(f={\widetilde{f}} ,f={\widehat{f}}\) are defined in \({\mathcal {G}}={\mathbb {B}}^{n}\) by ( 2.9), (2.11) in the cases \(\left| 2+k-4\lambda \right| \ge k\) and \(\left| 2+k-4\lambda \right| <k,\) respectively. The mappings \({\widetilde{F}},{\widehat{F}}\) belong to \({\mathcal {F}}_{1,k}\mathbf {(} {\mathbb {B}}^{n},{\mathbb {C}}^{n}\mathbb {)},\) by the enumerate above fact 2, because \({\widetilde{f}},{\widehat{f}}\in {\mathcal {F}}_{0,k}\mathbf {(}{\mathbb {B}} ^{n}\mathbf {,}\mathbb {C)}.\) Also \({\widetilde{F}},{\widehat{F}}\in S^{k} ({\mathbb {B}}^{n}),\) by the enumerated above fact l and the relations \({\widetilde{f}},{\widehat{f}}\in {\mathcal {M}}_{{\mathbb {B}}^{n}}^{k}.\)

First, we assume that \(\left| 2+k-4\lambda \right| \ge k,\) i.e., \(f={\widetilde{f}}.\) It is easy to check that in this case

$$\begin{aligned} Q_{f,k}=Q_{{\widetilde{f}},k}=\frac{2}{k}I^{k},\text { }Q_{f,2k}=Q_{\widetilde{f},2k}=\frac{k+2}{k^{2}}\left( I^{k}\right) ^{2}. \end{aligned}$$

For z \(=z^{0}\) and \(F={\widetilde{F}},\) i.e., \((f={\widetilde{f}}),\) we obtain (for the first below equality see the previous part of the proof)

$$\begin{aligned}&\left| \frac{T_{z}\left( D^{2k+1}F(0)(z^{2k+1})\right) }{\left( 2k+1\right) !\left\| z\right\| ^{2k+1}}-\lambda \left( \frac{T_{z}\left( D^{k+1}F(0)(z^{k+1})\right) }{\left( k+1\right) !\left\| z\right\| ^{k+1}}\right) ^{2}\right| \\&\quad =\frac{1}{\left\| z\right\| ^{2k}}\left| Q_{f,2k}(z)-\lambda \left( Q_{f,2k}(z)\right) ^{2} \right| \\&\quad =\frac{1}{\left\| z\right\| ^{2k}}\left| \frac{k+2}{k^{2}}\left( I^{k}(z)\right) ^{2}-\lambda \frac{4}{k^{2}}\left( I^{k}(z)\right) ^{2}\right| =\left| \left( I^{k}(\frac{z}{\left\| z\right\| })\right) ^{2}\right| \left| \frac{k+2}{k^{2}}-\lambda \frac{4}{k^{2} }\right| \\&\quad =\frac{1}{k}\max \left\{ 1,\frac{\left| 2+k-4\lambda \right| }{k}\right\} . \end{aligned}$$

Now, we assume that \(\left| 2+k-4\lambda \right| <k,\) i.e., \(f={\widehat{f}}.\) In this case it is easy to check that

$$\begin{aligned} Q_{f,k}=Q_{{\widehat{f}},k}=0,\text { }Q_{f,2k}=Q_{{\widehat{f}},2k}=\frac{1}{k}I^{2k}. \end{aligned}$$

For z \(=z^{0}\) and \(F={\widehat{F}},\) i.e., \((f={\widehat{f}}),\) we obtain similarly

$$\begin{aligned}&\left| \frac{T_{z}\left( D^{2k+1}F(0)(z^{2k+1})\right) }{\left( 2k+1\right) !\left\| z\right\| ^{2k+1}}-\lambda \left( \frac{T_{z}\left( D^{k+1}F(0)(z^{k+1})\right) }{\left( k+1\right) !\left\| z\right\| ^{k+1}}\right) ^{2}\right| =\frac{1}{\left\| z\right\| ^{2k}}\left| \frac{1}{k}I^{2k}(z)\right| \\&\quad =\frac{1}{k}\left| I^{2k}(\frac{z}{\left\| z\right\| })\right| =\max \left\{ 1,\frac{\left| 2+k-4\lambda \right| }{k}\right\} . \end{aligned}$$

\(\square \)

The identity (3.3) and Theorem 3.1 implies the following result of Fekete–Szegö type for starlike biholomorphic mappings in \({\mathbb {C}}^{n}\).

Theorem 3.2

For mappings \(F\in \widetilde{S^{*}}({\mathbb {B}}^{n})\) \(\cap {\mathcal {F}} _{1,k}\left( {\mathbb {B}}^{n},{\mathbb {C}}^{n}\right) ,k\in {\mathbb {N}}_{2},\) points \(z\in {\mathbb {B}}^{n}\backslash \left\{ 0\right\} \) and parameter \(\lambda \in {\mathbb {C}},\) there holds the following sharp estimate

$$\begin{aligned} \left| \frac{T_{z}\left( D^{2k+1}F(0)(z^{2k+1})\right) }{\left( 2k+1\right) !\left\| z\right\| ^{2k+1}}-\lambda \left( \frac{T_{z}\left( D^{k+1}F(0)(z^{k+1})\right) }{\left( k+1\right) !\left\| z\right\| ^{k+1}}\right) ^{2}\right| \le \frac{1}{k}\max \left\{ 1,\frac{\left| 2+k-4\lambda \right| }{k}\right\} , \end{aligned}$$

where \(T_{z}\in \left( {\mathbb {C}}^{n}\right) ^{*},z\in {\mathbb {B}} ^{n}\backslash \left\{ 0\right\} ,\) is arbitrary functional satisfying the conditions: \(\left\| T_{z}\right\| =1,\) \(T_{z}\left( z\right) =\left\| z\right\| .\)

In the other way a similar theorem was proved by Xu [30].

4 Final remarks

It is possible to allow also \(k=1\) in definition of \(\left( j,k\right) \)-symmetrical, \(j\in {\mathbb {Z}}\), functions, from \({\mathcal {F}}\left( {\mathcal {G}}\mathbf {,}{\mathbb {C}}^{m}\right) .\) Then \(\varepsilon =1\) and we should take the convention \({\mathcal {F}}_{j,1}({\mathcal {G}},{\mathbb {C}} ^{m})={\mathcal {F}}({\mathcal {G}},{\mathbb {C}}^{m})\) for \(j\in {\mathbb {Z}}.\) Consequently, in the case \(m=1\)

$$\begin{aligned}&{\mathcal {N}}_{{\mathcal {G}}}\cap {\mathcal {F}}_{0,1}\mathbf {(}{\mathcal {G}} \mathbf {,}\mathbb {C)=}{\mathcal {N}}_{{\mathcal {G}}},\\&\quad {\mathcal {M}}_{{\mathcal {G}}}\cap {\mathcal {F}}_{0,1}\mathbf {(}{\mathcal {G}} \mathbf {,}\mathbb {C)=}{\mathcal {M}}_{{\mathcal {G}}}={\mathcal {M}}_{{\mathcal {G}}} ^{1}={\mathcal {M}}_{{\mathcal {G}}}^{1}\cap {\mathcal {F}}_{0,1}\mathbf {(} {\mathcal {G}}\mathbf {,}\mathbb {C)}, \end{aligned}$$

while in the case \(m=n\)

$$\begin{aligned} \widetilde{S^{*}}({\mathbb {B}}^{n})\cap {\mathcal {F}}_{1,1}\left( {\mathbb {B}}^{n},{\mathbb {C}}^{n}\right) =\widetilde{S^{*}}({\mathbb {B}} ^{n})=\widetilde{S^{1}}({\mathbb {B}}^{n})=\widetilde{S^{1}}({\mathbb {B}}^{n} )\cap {\mathcal {F}}_{1,1}\left( {\mathbb {B}}^{n},{\mathbb {C}}^{n}\right) . \end{aligned}$$

Therefore, for \(\lambda \in {\mathbb {C}},\) we obtain the following sharp estimates

$$\begin{aligned} \mu _{{\mathcal {G}}}\left( Q_{f,2\;}-\lambda \left( Q_{f,1}\right) ^{2}\right)\le & {} \frac{1}{3}\max \left\{ 1,3\left| \lambda -1\right| \right\} ,\\ \mu _{{\mathcal {G}}}\left( Q_{f,2\;}-\lambda \left( Q_{f,1}\right) ^{2}\right)\le & {} \max \left\{ 1,\left| 3-4\lambda \right| \right\} ,\\ \left| \frac{T_{z}\left( D^{3}F(0)(z^{3})\right) }{3!\left\| z\right\| ^{3}}-\lambda \left( \frac{T_{z}\left( D^{2}F(0)(z^{2})\right) }{2!\left\| z\right\| ^{2}}\right) ^{2}\right|\le & {} \max \left\{ 1,\left| 3-4\lambda \right| \right\} ,z\in {\mathbb {B}}^{n} \backslash \left\{ 0\right\} , \end{aligned}$$

with \(T_{z}\in \left( {\mathbb {C}}^{n}\right) ^{*},\left\| T_{z} \right\| =1,T_{z}\left( z\right) =\left\| z\right\| ,\) for the families \({\mathcal {N}}_{{\mathcal {G}}},{\mathcal {M}}_{{\mathcal {G}}}={\mathcal {M}} _{{\mathcal {G}}}^{1},\widetilde{S^{*}}({\mathbb {B}}^{n})=\widetilde{S^{1} }({\mathbb {B}}^{n}),\) respectively.